Now lets look at a new model for solving equations, one that works in situations where backtracking doesn't.
The model of bags and blocks focuses on the idea of balance. Each bag holds an unknown number of blocks (eventually represented by a variable). Each bag in the same problem must hold the same number of blocks, just as a variable in an equation must stand for the same number every time it is used.

The challenge is to solve a series of problems by figuring out how many blocks are in each bag. Encourage participants to solve the problems by removing the same thing from each side of the scale, which is comparable to the more formal process of doing the same thing to both sides of an equation.

Problem C3 brings up the interesting question of what happens when the solution is not a whole number.

Gropus: Some might say that it's possible to change the model so that the blocks are made out of some material, like clay, that can be cut up into fractions of blocks. Others might say that the model cannot be used. There is no need for consensus on this issue -- the point is that when models fail, they can either be modified or abandoned for new methods.

Problem C5 revisits an idea from the previous session: equations that have no solution. We have seen this idea modeled as two parallel lines: no intersection means no solution. This is another way to think about the same situation. Taking an equal number of bags off each side leaves unequal numbers of individual blocks, which would not balance each other. Thus the original equation was never really in balance to begin with.

The representation of negative blocks in Problem C7 is even more difficult and is, in fact, a good example of when a model becomes more trouble than it's worth. Many models outgrow their usefulness in this way.